OPTIMAL MULTIPERIOD PORTFOLIO POLICIES the proportion a/A of initial wealth held mization problem, the level of initial in the risky asset. It is seen that da/da wealth has somehow slipped out the back 0; this is the disconcerting property men- door. Also, the resulting maximum level tioned above of any utility function ex- of expected utility would seem to be inde hibiting increasing absolute risk aver- pendent of initial wealth So it appears to be a conflict between With the optimal value of a given by the two formulations. A little reflection (5), maximum expected utility will be shows, however, that when initial wealth E2 is taken as a given, constant datum(say maxElL (Y)]=4a(v+E) 100), any level of final wealth can ob- (6)viously be equivalently described eitl in absolute terms(say, 120)or as a rate of return( 2). But Tobin's formulation.,s formu- wealth level of 120, it is immaterial to the lation is somewhat different 6 he also investor whether this is a result of assumes quadratic utility, but the argu- initial wealth of 80 with yield. 5 or an ment of the utility function is taken as initial wealth of 100 with yield. 2(or any one plus the portfolio rate of return. Sec- other combination of A and R such that ond, he takes as decision variable the AR= 120). The explanation of the ap- proportion of initial wealth invested in parent confict is now very simple: When the risky asset. If this fraction is called using a quadratic utility function in R, k, he thus wishes to maximize expected the coefficient p is not independent of A utility of the variateR= 1+kX. In the if the function shall lead to consistent symbols used above, decisions at different levels of wealth This is seen by obs Y-A+ax=1+x=1+kX. so that ervin thatR= Y/A, R Then with a quadratic utility function V(R)=R-BR', (7) which is equivalent, as a utility function, k is determined such that EV(R)] is a to Y-(B/A)Y. What this then, is that a utility function of the form R-βR2 cannot be used with the sameβ ELV(R)]= E[1+kX-B(1+kx] at different levels of initial wealth.The (1一)+(1-2)Ek appropriate value of B must be set such β(V+B)2 that B/A= athat is, B must be changed in proportion to A. But when An interior maximum is given by the this precaution is taken, Tobin's formu decision h(1-28)E lation will obviously lead to the correct (8)decision; with B= aA substituted in equation( 8), we get The important point to be made here is that the way( 8)is written, it seem k (1-2aA)E the optimal k is independent of 2aA(v+E2 weealth. In the formulation of the tha nat is (1-2aA)E 6 Tobin,“ Theory of Portfolio Selection.” his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and ConditionsOPTIMAL MULTIPERIOD PORTFOLIO POLICIES 217 the proportion a/A of initial wealth held in the risky asset. It is seen that da/dA < 0; this is the disconcerting property men￾tioned above of any utility function ex￾hibiting increasing absolute risk aver￾sion. With the optimal value of a given by (5), maximum expected utility will be maxE[U(Y)] =4a(V+E2) (6) + V (A- A ) V+E 2 Tobin's formulation.-Tobin's formu￾lation is somewhat different.6 He also assumes quadratic utility, but the argu￾ment of the utility function is taken as one plus the portfolio rate of return. Sec￾ond, he takes as decision variable the proportion of initial wealth invested in the risky asset. If this fraction is called k, he thus wishes to maximize expected utility of the variate R = 1 + kX. In the symbols used above, Y A+aX=1+ a X=+kX. A A A Then with a quadratic utility function V(R)=R- 3R2, (7) k is determined such that E[V(R)] is a maximum: E[V(R)] = E[1 + kX - (1 + kX)2] = (1 - A) + (1 - 23)Ek - 3(V + E2)k2 . An interior maximum is given by the decision (1- 2f3)E The important point to be made here is that the way (8) is written, it seems as if the optimal k is independent of initial wealth. In the formulation of the maxi￾mization problem, the level of initial wealth has somehow slipped out the back door. Also, the resulting maximum level of expected utility would seem to be inde￾pendent of initial wealth. So it appears to be a conflict between the two formulations. A little reflection shows, however, that when initial wealth is taken as a given, constant datum (say, 100), any level of final wealth can ob￾viously be equivalently described either in absolute terms (say, 120) or as a rate of return (.2). But in considering a final wealth level of 120, it is immaterial to the investor whether this is a result of an initial wealth of 80 with yield .5 or an initial wealth of 100 with yield .2 (or any other combination of A and R such that AR = 120). The explanation of the ap￾parent conflict is now very simple: When using a quadratic utility function in R, the coefficient f is not independent of A if the function shall lead to consistent decisions at different levels of wealth. This is seen by observing that R = Y/A, so that V(R) =V = _: I which is equivalent, as a utility function, to Y - (3/A) Y2. What this implies, then, is that a utility function of the form R - OR2 cannot be used with the same 13 at different levels of initial wealth. The appropriate value of : must be set such that 13/A = a-that is, : must be changed in proportion to A. But when this precaution is taken, Tobin's formu￾lation will obviously lead to the correct decision; with A = aA substituted in equation (8), we get a (1-2aA)E A 2aA (V+E2)' that is, ( 1-2aA)E a 2a(V+E2) 6 Tobin, "Theory of Portfolio Selection." This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions